CHAP. V] GENERALIZATIONS OF EULER'S ^-FUNCTION. 151 yields or which is next to the last formula of Gegenbauer's.204 Similarly, which is the case t = l of Gegenbauer's72 fifth formula in Ch. X, ak(n) being the sum of the &th powers of the divisors of n. E. Weyr211 interpreted J2(n) in connection with involutions on loci of genus 1. From the same standpoint, L. Gegenbauer212 proved (12) for k = 2 and noted that the value (10) of J2 (n) then follows by the usual method of number-theoretic derivatives. L. Gegenbauer2120 wrote <j>k(m, n) for the number of sets of k positive integers ^ra whose g. c. d. is prime to w = p1V . .p^ and proved a formula including M* =&(m, n) + i £ (Xx, . . . X)2 where (Xi, . . . , Xff) is the determinant derived from that with unity throughout the main diagonal and zeros elsewhere by replacing the 7th row by the X7th row for 7 = !,. . ., a. The case m = n, k = l, is due to Pepin.37 There is an analogous formula involving the sum of the fcth powers of the positive integers ^m and prime to n. E. Jablonski74 used Jk(n) in connection with permutations. G. Arnoux213 proved (10) in connection with modular space. *J. J. Tschistiakow214 (or Cistiakov) treated the function /&(n). R. D. von Sterneck215 proved that Jk(n) =S/r(A1)A-r(X2) =2^>(X1) . . .<KA*), the X's ranging over all sets of integers ^ n whose 1. c. m. is n. To generalize this, let Jk(nm, mlj . . . , mk) be the number of sets of integers i1}. . . , ik, whose g. c. d. is prime to n, while i^n/m,- for j = 1, . . . , k. Then Jk(n] mi,..., raJ=2Jr(Ai; m'lt. . ., m'r)JA_r(X2; m'r+l,. . ., m'k) =2/1(X1; mO . . . Ji(\k] mk), the X's ranging over all sets of integers ^n whose 1. c. m. is n, while rn'i,. . . , m'k form any fixed permutation of wb . . . , mk, and Ji(n; m), designated <j>(m)(n) by the author, is the number of integers ^n/m which are prime to n. Also, 211Sitzungsberichte Ak. Wiss. Wicn (Math.), 101, Ha, 1892, 1729-1741. 212Monatshefte Math. Phys., 4, 1893, 830. 212aDenkschr. Ak. Wiss. Wien (Math.), 60, 1893, 25-47. 213Arithm6tiquo graphique; espaccs arith. hyperrnagiqu.es, 1894, 93. 214Math. Soc. Moscow, 17, 1894, 530-7 (in Russian). '"Monatshefte Math. Phys., 5, 1894, 255-266.